Questions: Fraunhofer Diffraction: Far-Field Diffraction Patterns

Question 1

A laser shines through a narrow slit (0.1 mm wide) and then through a wide slit (1 mm wide) in two separate experiments. Compared to the wide slit, the narrow slit produces a central diffraction maximum that is:

• A. Narrower, because less light passes through and so it spreads less
• B. The same width, because wavelength determines the pattern, not slit width
• C. Wider, because localizing the wave spatially spreads it more in angle
• D. Brighter but the same width, because the smaller aperture concentrates the beam

Answer: (C)

This is the Fourier reciprocity principle: a narrower slit (smaller a) means the function describing the aperture is more spatially confined, and its Fourier transform is broader in angular frequency. The first minimum occurs at sin θ = λ/a — a smaller 'a' gives a larger θ. This is a spatial analogue of the Heisenberg uncertainty principle. The intuitive but wrong answer is option A — students often expect a narrow opening to produce a narrow beam, as if light were a particle stream.

Question 2

The Fraunhofer diffraction pattern produced by a complex aperture is mathematically equivalent to:

• A. The convolution of the aperture function with the wavelength
• B. The magnitude-squared of the Fourier transform of the aperture function
• C. The integral of the aperture function divided by the observation distance
• D. The autocorrelation of the aperture function multiplied by wavelength

Answer: (B)

In the far-field limit, the amplitude at angle θ is the Fourier transform of the aperture function evaluated at spatial frequency sin θ/λ. The observed intensity is the square of the amplitude, giving the magnitude-squared of the Fourier transform. This is why diffraction gratings (periodic apertures) produce sharp intensity peaks — the Fourier transform of a periodic comb function is another comb function.

Question 3

A larger telescope aperture produces a narrower diffraction-limited angular resolution.

• T. True
• F. False

Answer: True

The angular radius of the Airy disk (the central diffraction maximum for a circular aperture) is proportional to λ/D, where D is the aperture diameter. A larger aperture decreases this angle, allowing finer angular details to be resolved. This is why large telescopes can distinguish stars that appear as a single point through smaller instruments, and why radio telescopes must be enormous — radio wavelengths are centimeters to meters, requiring proportionally huge apertures for the same angular resolution as optical telescopes.

Question 4

In a single-slit Fraunhofer diffraction pattern, the secondary maxima are the same width as the central maximum.

• T. True
• F. False

Answer: False

The central maximum is twice as wide as each secondary maximum — it extends from −λ/a to +λ/a in sin θ, while each secondary maximum spans only λ/a. This double-width central fringe is a distinctive signature of single-slit Fraunhofer diffraction. The asymmetry arises because the central maximum spans the range between the ±1st-order minima (on both sides), while secondary maxima each span the range between two adjacent minima on the same side.

Question 5

Why does narrowing a slit produce a wider diffraction pattern, rather than a narrower, more beam-like pattern?

Model answer: Because the diffraction pattern is the Fourier transform of the aperture, and Fourier transforms obey a reciprocal relationship: spatial confinement in one domain produces spreading in the reciprocal domain. A narrower slit confines the wavefront more tightly in position, which — by the uncertainty principle for waves — requires a broader distribution of transverse momenta (directions). Each point in the slit emits a Huygens wavelet in all directions, and with fewer points across the slit, there are fewer opportunities for destructive interference to cancel light at large angles, so light spreads more widely.

The intuition that 'a smaller hole makes a smaller spot' is the particle-optics approximation, valid when the wavelength is much smaller than the slit. When the slit width approaches the wavelength, wave effects dominate and the reciprocal relationship asserts itself. This same principle governs why a single atom can diffract electrons, why X-ray diffraction patterns reveal atomic spacings, and why aperture size is the fundamental limit on telescope resolution.